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The CM system

The dynamical system that is generally called Calogero-Moser (hereafter CM) is the model characterized by the Hamiltonian\[\tag{1}
H\left( \underline{p},\underline{q}\right) =\frac{1}{2}\sum_{n=1}^{N}\left(p_{n}^{2}+\omega ^{2}q_{n}^{2}\right) +g^{2}\sum_{m,n=1;m\neq n}^{N}\left(q_{n}-q_{m}\right) ^{-2}~.
\]

It has the merit – in contrast to (1) – of being translation-invariant, and yet to yield, in its center-of-mass system, an almost identical dynamics to that yielded by the Hamiltonian (1): the difference among the two models is that the center-of-mass oscillates harmonically in the first case, (1), while it moves freely in the second case, (2).

The interest of this model lies in its exact solvability, both in the classical and quantal contexts.

In the classical context, the Hamiltonian model (1) is completely integrable – \(N\) integrals of motion in involution can be explicitly exhibited – indeed superintegrable – \(2N-1\) functionally independent integrals of motion can be explicitly exhibited – and algebraically solvable: the solution of the initial-value problem of the Newtonian equations of motion (3) can be performed
by algebraic operations, specifically by computing the \(N\) eigenvalues of an \(N\times N\) matrix explicitly known in terms of the initial data \(q_{n}\left( 0\right) \ ,\) \(\dot{q}_{n}\left( 0\right) \) and of the time \(t\ .\) In the confined case (\(\omega >0\)), the solution is isochronous: completely periodic,
\[
q_{n}\left( t+T\right) =q_{n}(t)~,~~~n=1,...,N~,
\]
for arbitrary initial data, with the fixed period
\[\tag{5}
T=\frac{2\pi }{\omega }~.
\]

In the not confined case (\(\omega =0\)) the time evolution of the \(N\) particles features the following neat asymptotic relation among their positions and velocities \(p_{n}=\dot{q}_{n}\ ,\) in the remote past and future:
\[\tag{6}
q_{n}\left( t\right) =p_{n}^{\left( \pm \right) }\,t+q_{n}^{\left( \pm \right) }+O\left( t^{-1}\right) ~~~\text{as}~~~t\rightarrow \pm \infty ~,
\]

with the ordering of these velocities corresponding of course to the ordering of the particles on the line, say \(p_{1}^{\left( -\right)}>p_{2}^{\left( -\right) }>...>p_{N}^{\left( -\right) }\) and \(p_{1}^{\left(
+\right) }<p_{2}^{\left( +\right) }<...<p_{N}^{\left( +\right) }\) corresponding to \(q_{1}\left( t\right) <q_{2}\left( t\right) <...<q_{N}\left( t\right) \ :\) note that the ordering of the particles on the
line cannot change throughout the time evolution due to the singular nature of the two-body interaction at zero separation, forbidding them from overtaking each other. This outcome, (6), is independent of the
value of \(g^{2}\) (provided \(g^{2}>0\)). It is congruent with what happens for extremely small \(g^{2}\) when the particles move freely except for colliding elastically; but of course for arbitrary (nonvanishing) \(g^{2}\) the motion of each particle is influenced by the motion of every other particle, indeed significantly so since this is a many-body problem with long-range two-body interactions. It is indeed instructive to try and make for oneself a graph of the positions \(q_{n}\left( t\right) \) of the \(N\) particles as functions of the time \(t\) reflecting the features described above, for small
and large values of \(g^{2}\ .\)

In the quantal context (which, remarkably, was solved firstly: see below), in the confined case (\(\omega ^{2}>0\)) the spectrum of the stationary Schrödinger equation (4) is discrete and equispaced, being given by the simple formula
\[\tag{8}
E_{k}=\omega \left[ k+\frac{N}{2}+\frac{N\,\left( N-1\right) }{2}\left( a+\frac{1}{2}\right) \right] ~~~~,~k=0,1,2,...~,
\]

\[
a=\frac{1}{2}\left( 1+2g^{2}\right) ^{1/2}~,
\]
with every energy level having the same degeneracy as in the pure oscillator potential (\(g^{2}=0\)) with Fermi statistics. For \(g^{2}=0\) hence \(a=1/2\) this coincides indeed with the spectrum of the pure oscillator model with Fermi statistics – consistently with the vanishing, for \(g^{2}>0\) (hence
also in the limit \(g^{2}\rightarrow 0^{+}\)), of the wave function \(\Psi\left( x_{1},...,x_{N}\right) \) whenever two coordinates coincide, due to the singularly repulsive character of the two-body force at zero separation. For \(a=-1/2\) the spectrum (8) coincides with that of the pure
oscillator (\(g^{2}=0\)) case with Boltzmann, or Bose, statistics. In the distinguishable particles (Boltzmann statistics) case the fixed ordering of the particles on the line in the classical CM model
(with \(g^{2}>0\)) corresponds of course in the quantal context to the vanishing of the wave function \(\Psi \left( x_{1},...,x_{N}\right) \) for all orderings of the coordinates \(x_{n}\) except a specific one, say \(\Psi \left( x_{1},...,x_{N}\right) =0\) unless \(x_{1}<x_{2}<...x_{N}\ ;\) for identical particles (with Bose or Fermi statistics; or even for anions) the eigenfunction shall be continued in the other sectors as dictated by the statistics – with negligible effects on the dynamics, since the particles
are effectively distinguishable via their ordering. The degeneracy of the spectrum (for identical particles, Bose or Fermi) can be displayed by replacing \(k\) in (8) with \(\sum_{j=1}^{N}jn_{j}\ ,\) where each of the \(N\) quantum numbers \(n_{j}\) can be a nonnegative integer and a
different eigenfunction corresponds to every different set of \(N\) numbers \(n_{j}\) (in the Boltzmann case there is an additional degeneracy \(N!\) of every energy level).

In the not confined case (\(\omega =0\)), the scattering matrix can be computed in closed form, and it vanishes unless the initial and final momenta, \(p_{n}\) and \(p_{n}^{\prime },\) satisfy the restrictions, \(p_{n}^{\prime }=p_{N+1-n}\ ,\) corresponding to the classical outcome, see (6) ("no diffraction").

The solvable character of the CM model is relevant in a physical context inasmuch as it provides a toy many-body problem with interparticle pair forces singularly repulsive at short-distance (as are the forces in typical, molecular or nuclear, many-body problems), both when the system is confined due to the external harmonic oscillator potential (\(\omega ^{2}>0\)) – mimicking a container, although one with soft walls – and as well when there is no containment (\(\omega^{2}=0\)) and the key physical phenomena are then scattering processes (in the freely moving center-of-mass system). The main shortcoming with respect to physical reality is the one-dimensional character of this model, and the special form of the two-body interaction, causing the peculiarly neat phenomenology outlined above. Indeed – rather than as a toy physical many-body problem (although originally invented with this application in mind) – this model turned out to be more relevant in other areas of theoretical and
mathematical physics and in pure mathematics: as one of the first nontrivial examples of integrable dynamical system, and because of its connection with various (old and new) developments in several fields. (Two assessments: "The last decade has witnessed a true explosion of activities involving Calogero-Moser-Sutherland models, and these now play a role in research areas ranging from theoretical physics (such as soliton theory, quantum field theory, string theory, solvable models of statistical mechanics, condensed matter physics, and quantum chaos) to pure mathematics (such as representation theory, harmonic analysis, theory of special functions, combinatorics of symmetric functions, dynamical systems, random matrix theory, and complex geometry)" [see(vDV2000)]; "Calogero-Moser
systems, which were originally discovered by specialists in integrable systems, are currently at the crossroads of many areas of mathematics and within the scope of interests of many mathematicians. More specifically, these systems and their generalizations turned out to have intrinsic connections with such fields as algebraic geometry (Hilbert schemes of surfaces), representation theory (double affine Hecke algebras, Lie groups, quantum groups), deformation theory (symplectic reflection algebras), homological algebra (Koszul algebras), Poisson geometry, etc." [see(E2005)]. Digiting Calogero Moser on Google (Monday 19 June and Thursday 29 June, 2008) yielded (approximately) 40,600 and 41,000 links; Calogero-Moser, 37,000 and 37,500; Calogero Moser system, 26,900 and 27,300; Calogero Moser dynamical system, 10,500 and 15,700; Calogero Sutherland, 40,400 and 41,700 (whatever this means: Moser Sutherland yielded 157,000 and 148,000 links, but the second and fourth were a YouTube site with a video featuring both (Edda) Moser and (Joan) Sutherland singing a Mozart arietta...).

Additional results

There are several variations on, and generalizations of, the CM system, and connections with other problems (in addition to those mentioned above). In this section they are outlined quite tersely: the reader whose interest is
stimulated by these hints will have no difficulty to pursue the specific matter via the web (possibly also using as guidance some of the names quoted below), or by consulting the textbooks listed at the end of the following section.

and may be considered a more appropriate toy model for the nuclear many-body problem, inasmuch as the pair potential is now short-range. Of course for \(a=0\) both these Hamiltonians reduce back to the CM Hamiltonian
(1).

A more general extension of the CM model (with \(\omega =0\)) is characterized by the Hamiltonian
\[
H\left( \underline{p},\underline{q}\right) =\frac{1}{2} \sum_{n=1}^{N}p_{n}^{2}+g^{2}\,a^{2}\sum_{m,n=1;m\neq n}^{N}\wp \left[a\left( q_{n}-q_{m}\right) ;\omega _{1},\omega _{2}\right] ~,
\]
where \(\wp \left( z;\omega _{1},\omega _{2}\right) \) is the Weierstrass function, namely the elliptic function with a double pole at \(z=0\ ,\) doubly-periodic in the complex \(z\)-plane with semi-periods \(\omega_{1},~\omega _{2}\) (standard assignment: one of them real and the other imaginary, yielding a Weierstrass function \(\wp \left(z;\omega _{1},\omega _{2}\right) \) that is real when \(z\) is real). This reduces (up to trivial renormalizations) to the Sutherland Hamiltonian (9) if \(\omega _{1}=\infty ,~\omega _{2}=i\pi /2\ ;\) it also reduces to the CM model (1) (with \(\omega =0\)) in the limit of vanishing \(a\ ,\) or if both semiperiods diverge.

In the classical context all these Hamiltonian models are integrable: they are also solvable, at least in the trigonometric and hyperbolic cases. In the trigonometric ("Sutherland") case all motions are multiply periodic (all solutions are a linear superposition of \(N\) trigonometric functions the periods of which depend on the initial data). In the hyperbolic case the asymptotic outcome (6) obtains again, but now with (7) replaced by
\[
q_{n}^{\left( +\right) }=q_{N+1-n}^{\left( -\right) }+\sum_{m=1;m\neq n}^{N}\Delta \left( p_{m}^{\left( -\right) }-p_{n}^{\left( -\right) };g;a\right) ~,
\]
\[
\Delta \left( p;g;a\right) =\frac{\text{sign}\left( p\right) }{2a}\,\log \left[ 1+\left( \frac{ga}{p}\right) ^{2}\right] ~.
\]
These formulas refer of course to the same particle ordering on the line as in (6), and they clearly imply that, in the remote future, the asymptotic shift among the position of the \(n\)-th particle, and what the position of the \(\left( N+1-n\right) \)-th particle would have been in the absence of any interaction, amounts just to the sum of the \(N-1\) shifts that would be due to a sequence of its "only two-body" interaction – one by one – with each of the other \(N-1\) particles, without the remaining ones being
present; indeed – as implied by these very formula in the \(N=2\) case – \(\Delta \left( p_{1}-p_{2};g;a\right) \) is such a shift experienced by particle 1 due to its interaction with particle 2, its sign depending on the relative velocity of the two particles (likewise \(\Delta \left(
p_{2}-p_{1};g;a\right) =-\Delta \left( p_{1}-p_{2};g;a\right) \) is the shift experienced by particle 2). This "factorization property" holds, in spite of the fact that throughout the motion every particle is influenced by its
interaction with all the other \(N-1\) particles. In the elliptic case the complete solution of the problem is considerably more complicated; key contributions to obtain it have been made by Igor Krichever and by
Vladimir Inozemtsev.

In the quantal context the spectrum is discrete (a quadratic function of the integer quantum numbers) in the trigonometric case (9) ("no bands", in spite of the periodic character of the potential), and the scattering matrix is very simple (again, "no diffraction") in the hyperbolic case (10). The treatment of the quantal elliptic case is, again, much more complicated: key contributions have been made by many researchers including Edwin Langmann and (also in a more general context, see below) Simon Ruijsenaars and Felipe van Diejen.

Another type of extension consisted in the insertion of more general "external potentials" than the simple harmonic one featured by the Hamiltonian (1); the more general such potential compatible with the preservation of the integrable character of the model is a quartic polynomial, with arbitrary coefficients.

Another type of generalization led to the treatment of more than one type of particles: for instance a (still exactly treatable) generalization of the model (10) – obviously of some interest as a toy many-body problem
– includes arbitrary numbers of equal mass particles of two types, with the repulsive pair potential \(g^{2}a^{2}/\sinh ^{2}\left( ax\right) \) acting among equal particle, and the attractive
potential \(-g^{2}a^{2}/\cosh ^{2}\left( ax\right) \) acting among different particles – hence also giving rise to two-body bound states ("molecules"), with the possibility to investigate exactly the phenomenology
of interactions among all such objects – both in the classical and quantal contexts.

Interesting connections have been uncovered (by Martin Kruskal, Jürgen Moser and others) among the motions of the particles described, in the classical context, by the Newtonian equations of motion (3) – but with the coordinates \(q_{n}=z_{n}\) considered as complex numbers – and the time evolution of the poles of certain solutions of the Korteweg-de Vries PDE (prototypical integrable nonlinear evolution equation in
1+1 variables: one-dimensional space and time); likewise interesting connections have also been uncovered (mainly by Igor Krichever) among CM-type models and the Kadomtsev-Petviashvili equation (prototypical integrable nonlinear evolution PDE in 2+1 variables). The connection among the positions \(z_{n}\left( t\right) \) of particles evolving according to integrable many-body models and the poles or zeros of the solution \(\psi (z,t)\) of (integrable nonlinear, or linear) PDEs has indeed been an important tool to identify several (classes of) new integrable many-body problems and to investigate their time evolutions [see(C1978)].

Another line of generalization has associated internal degrees of freedom ("spin") to the particles whose time evolution is described by CM-type models.

An extension that led to many mathematical developments emerged from the replacement of the quantity \(q_{n}-q_{m}\) in the above equations (see for instance (2)) with \(\underline{\alpha }\cdot \underline{q}\) where the \(N\)-vector \(\underline{\alpha }\) characterizes a "root system".

An extremely interesting extension of the CM system (including its trigonometric, hyperbolic and elliptic generalizations, see above) has resulted in a class of RS models (see below), it would probably be better to specify it here </review> that can be considered as a "relativistic" generalization of it – to the extent this is possible for many-body models describing evolutions as functions of an independent variable interpreted as (absolute) time. This RS extension features an additional constant, playing the role of "speed of light", and reproduces the CM models when this constant is set to infinity. The justification for considering this a relativistic extension is because the Galilei invariance of the CM model gets replaced by the Poincaré invariance of the extended model.

Another class of extensions obtains from the replacement – in the classical context – of the Hamiltonian (1) with
\[
H\left( \underline{p},\underline{z}\right) =\frac{1}{2}\sum_{n=1}^{N}\left(
p_{n}^{2}+\omega ^{2}z_{n}^{2}\right) +\sum_{m,n=1;m\neq n}^{N}g_{nm}^{2}\,\left( z_{n}-z_{m}\right) ^{-2}~.
\]
Here we replaced again the canonical coordinates \(q_{n}\) with \(z_{n}\) to emphasize that these coordinates, as well of course as the corresponding canonical momenta \(p_{n}\ ,\) shall now be considered as complex variables (namely, the motions \(z_{n}\equiv z_{n}\left( t\right) \) take now place in the complex \(z\)-plane; the time \(t\) is of course always real). The generalization is due to the fact that there are now different coupling constants (possibly also complex) for every particle pair. This generally destroys the integrable character of this Hamiltonian system; but it has been shown that this model is
nevertheless isochronous, in the following sense: there is an open, fully-dimensional region in the (complex) phase space of this system where all motions are completely periodic (periodic in all degrees of freedom) with period \(T,\) see (5). In other regions of phase space the motions can also be periodic, but with periods that are integer (possibly quite large) multiples of \(T\ ;\) or they can be aperiodic, in some sense chaotic (at least in the sense of displaying a sensitive dependence
on their initial data) [see(CS2002)].

Finally, as we just indicated, it is possible – and generally appropriate and interesting – to extend the study of all the models mentioned above from the real to the complex, by considering the motion of the "particles", identified by their coordinates \(z_{n}\left( t\right) ,\) to take place in the complex \(z\)-plane rather than on the real line. It is thereby also possible to go from real one-dimensional models ("motions on the line") to real two-dimensional models ("motions in the plane"). And it is even possible to do so compatibly with some kind of rotation-invariant (or at least covariant, in terms of two-vectors) character for these latter models. The models thereby obtained are interesting inasmuch as they are no more constrained by the time-independent ordering that characterizes the standard CM model on the line: particles can now go around each other. Hence this extension generally yields many-body models, whose Hamiltonian and solvable characters are preserved, but which feature richer dynamics.

Historical notes

The first paper introducing and solving, in a quantal context, the CM system on the line with arbitrary \(N\) is [C1971] (for \(N=3\) this problem had been solved a bit earlier [see(C1969)]; the analogous model on the circle – or equivalently with the rational (inverse-square) potential in (1) replaced by the trigonometric (inverse-sine-square) potential, see (9) – was treated, essentially simultaneously and
certainly independently, by Bill Sutherland [cf.(S1971)]. Jürgen Moser [cf.(M1975)] proved that the CM model is, in the classical context, integrable indeed solvable, for arbitrary \(N\) (for \(N=3\)
this problem had been solved somewhat earlier by Carlo Marchioro, and to some extent – much earlier – by Carl Jacobi [cf.(J1866)] – although Jacobi's contribution had been forgotten and was only rediscovered later, by Askol'd Perelomov). Legend has it that Moser was initially sceptical about the conjectures on the classical behavior of the CM system proffered in [C1971]: his input was then instrumental to prove their validity. His contribution fitted into the bloom of investigations on integrable systems following the discovery by Morikazu Toda of the first instance of integrable one-dimensional many-body problem (with nearest-neighbor exponential interactions) [cf.(T1967)] and by Martin Kruskal and his collaborators of the solvable character of the Korteweg-de Vries equation [cf.(GGKM1967)], and profited (as most of the subsequent investigations) from the fundamental contribution by Peter Lax (the notion of "Lax pairs" [cf.(L1968)]). The early papers related to the CM model began to label it as "Calogero Moser" or (especially in the trigonometric variant) by using the label "Sutherland" (alone or in conjunction with other names). The extension to elliptic interactions was introduced, in the classical context, in [C1975]; this involved the introduction and solution of certain functional equations, which subsequently gave rise to a mathematical trend of its own [cf.(BC1990)]. In this same paper [cf.(C1975)] the model featuring particles of different types (see above) was also introduced. The more general results on the CM model with an external (in fact quartic) potential are due to Vladimir Inozemtsev and Stefan Wojciechowski. The original idea of the extension to other root systems is due to Misha Olshanetsky and Askol'd Perelomov, who also wrote two seminal review papers on the classical and quantal aspects of
these models [cf.(OP1981), (OP1983)]; many contributed to subsequent developments, including Ryu Sasaki. Another seminal paper for mathematical developments was (KKS1978). The generalization from CM to RS ("Ruijsenaars-Schneider" or "relativistic") many-body models was introduced in [RS1986] and pursued by Simon Ruijsenaars and others in many subsequent papers. For the treatment of isochronous systems see the book with this title listed below.

Apologies are offered to all those who made important contributions to developments connected with the CM model and are not mentioned here: they are too many to fit into a Scholarpedia entry.

Acknowledgements

It is a pleasure to thank Mario Bruschi, Jean-Pierre Françoise and
Matteo Sommacal for having accepted to be co-curators of this Scholarpedia
entry and for helpful feedbacks on this article.